As is well known, the following theory is equiconsistent with $PA$:

$ZFC$ with the axiom of infinity replaced by its negation.

Since this theory is equiconsistent with $PA$, it would seem reasonable to infer (wouldn't it?) that the consistency of '$ZFC$ with the axiom of infinity replaced by its negation' could be provable in "$PRA$ + '_ quantifier-free transfinite induction up to $\epsilon_0$_".

So what 'theory of sets'(?) is equiconsistent with "$PRA$ + '_quantifier- free transfinite induction up to $\epsilon_0$"?

  Also, can one define a notion of forcing in the aforementioned theory?

(If this seems too silly a question, please let me know and I will delete....)

As is well known, the following theory is equiconsistent with $PA$:

$ZFC$ with the axiom of infinity replaced by its negation.

Since this theory is equiconsistent with $PA$, it would seem reasonable to infer (wouldn't it?) that the consistency of '$ZFC$ with the axiom of infinity replaced by its negation' could be provable in "$PRA$ + $TI({\epsilon_0})$.

So what 'theory of sets'(?) is mutually interpretable with "$PRA$ + $TI({\epsilon_0})$? Also, can one define a notion of forcing in the aforementioned theory?

(If this seems too silly a question, please let me know and I will delete....)

As is well known, the following fragment of $ZFC$ is equiconsistent with $PA$:

$ZFC$ with the axiom of infinity replaced by its negation.

Since this fragment of $ZFC$ is equiconsistent with $PA$, it would seem reasonable to infer (wouldn't it?) that the consistency of '$ZFC$ with the axiom of infinity replaced by its negation' could be provable in "$PRA$ + '_ quantifier-free transfinite induction up to $\epsilon_0$_".

So what fragment of $ZFC$ is equiconsistent with "$PRA$ + '_quantifier- free transfinite induction up to $\epsilon_0$"?

Also, can one define a notion of forcing in the aforementioned fragment?

(If this seems too silly a question, please let me know and I will delete....)

As is well known, the following theory is equiconsistent with $PA$:

$ZFC$ with the axiom of infinity replaced by its negation.

Since this theory is equiconsistent with $PA$, it would seem reasonable to infer (wouldn't it?) that the consistency of '$ZFC$ with the axiom of infinity replaced by its negation' could be provable in "$PRA$ + '_ quantifier-free transfinite induction up to $\epsilon_0$_".

So what 'theory of sets'(?) is equiconsistent with "$PRA$ + '_quantifier- free transfinite induction up to $\epsilon_0$"?

Also, can one define a notion of forcing in the aforementioned theory?

(If this seems too silly a question, please let me know and I will delete....)